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Mirrors > Home > MPE Home > Th. List > idghm | Structured version Visualization version GIF version |
Description: The identity homomorphism on a group. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
Ref | Expression |
---|---|
idghm.b | ⊢ 𝐵 = (Base‘𝐺) |
Ref | Expression |
---|---|
idghm | ⊢ (𝐺 ∈ Grp → ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Grp) | |
2 | 1 | ancli 574 | . 2 ⊢ (𝐺 ∈ Grp → (𝐺 ∈ Grp ∧ 𝐺 ∈ Grp)) |
3 | idghm.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺) | |
4 | eqid 2622 | . . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
5 | 3, 4 | grpcl 17430 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝐺)𝑏) ∈ 𝐵) |
6 | 5 | 3expb 1266 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝐺)𝑏) ∈ 𝐵) |
7 | fvresi 6439 | . . . . . 6 ⊢ ((𝑎(+g‘𝐺)𝑏) ∈ 𝐵 → (( I ↾ 𝐵)‘(𝑎(+g‘𝐺)𝑏)) = (𝑎(+g‘𝐺)𝑏)) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( I ↾ 𝐵)‘(𝑎(+g‘𝐺)𝑏)) = (𝑎(+g‘𝐺)𝑏)) |
9 | fvresi 6439 | . . . . . . 7 ⊢ (𝑎 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑎) = 𝑎) | |
10 | fvresi 6439 | . . . . . . 7 ⊢ (𝑏 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑏) = 𝑏) | |
11 | 9, 10 | oveqan12d 6669 | . . . . . 6 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → ((( I ↾ 𝐵)‘𝑎)(+g‘𝐺)(( I ↾ 𝐵)‘𝑏)) = (𝑎(+g‘𝐺)𝑏)) |
12 | 11 | adantl 482 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((( I ↾ 𝐵)‘𝑎)(+g‘𝐺)(( I ↾ 𝐵)‘𝑏)) = (𝑎(+g‘𝐺)𝑏)) |
13 | 8, 12 | eqtr4d 2659 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( I ↾ 𝐵)‘(𝑎(+g‘𝐺)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝐺)(( I ↾ 𝐵)‘𝑏))) |
14 | 13 | ralrimivva 2971 | . . 3 ⊢ (𝐺 ∈ Grp → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( I ↾ 𝐵)‘(𝑎(+g‘𝐺)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝐺)(( I ↾ 𝐵)‘𝑏))) |
15 | f1oi 6174 | . . . 4 ⊢ ( I ↾ 𝐵):𝐵–1-1-onto→𝐵 | |
16 | f1of 6137 | . . . 4 ⊢ (( I ↾ 𝐵):𝐵–1-1-onto→𝐵 → ( I ↾ 𝐵):𝐵⟶𝐵) | |
17 | 15, 16 | ax-mp 5 | . . 3 ⊢ ( I ↾ 𝐵):𝐵⟶𝐵 |
18 | 14, 17 | jctil 560 | . 2 ⊢ (𝐺 ∈ Grp → (( I ↾ 𝐵):𝐵⟶𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( I ↾ 𝐵)‘(𝑎(+g‘𝐺)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝐺)(( I ↾ 𝐵)‘𝑏)))) |
19 | 3, 3, 4, 4 | isghm 17660 | . 2 ⊢ (( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ Grp) ∧ (( I ↾ 𝐵):𝐵⟶𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( I ↾ 𝐵)‘(𝑎(+g‘𝐺)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝐺)(( I ↾ 𝐵)‘𝑏))))) |
20 | 2, 18, 19 | sylanbrc 698 | 1 ⊢ (𝐺 ∈ Grp → ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 I cid 5023 ↾ cres 5116 ⟶wf 5884 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 Grpcgrp 17422 GrpHom cghm 17657 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-ghm 17658 |
This theorem is referenced by: gicref 17713 symgga 17826 0frgp 18192 idrhm 18731 idlmhm 19041 frgpcyg 19922 nmoid 22546 idnghm 22547 idrnghm 41908 |
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